A second order linear homogeneous ordinary differential equation with constant coefficients can be expressed as

This equation implies that the solution is a function whose derivatives keep the same form as the function itself and do not explicitly contain the independent variable , since constant coefficients are not capable of correcting any irregular formats or extra variables. An elementary function which satisfies this restriction is the exponential function .

Substitute the exponential function into the above differential equation, the characteristic equation of this differential equation is obtained